Algebra Functions and Graphing

Study Notes

Domain and Range:
- Domain: set of all input values (x) for which the function is defined
- Range: set of all output values (y) that the function can produce
Function Operations:
- Sum/Difference: (f ± g)(x) = f(x) ± g(x)
- Product: (fg)(x) = f(x) * g(x)
- Composition: (f ∘ g)(x) = f(g(x))
Function Properties:
- Even/Odd: f(-x) = f(x) or f(-x) = -f(x)
- Increasing/Decreasing: f(x) ≥ f(y) or f(x) ≤ f(y) for x > y

Graph Types:
- Linear: straight line, y = mx + b
- Quadratic: parabola, y = ax^2 + bx + c
- Exponential: y = a * b^x
- Trigonometric: y = a * sin(x) or y = a * cos(x)
Graph Transformations:
- Vertical Shift: y = f(x) ± k
- Horizontal Shift: y = f(x ± h)
- Reflection: y = -f(x) or y = f(-x)
Graphing Techniques:
- Plotting points
- Using symmetry
- Identifying asymptotes

Angles and Triangles:
- Degree measure: 0° ≤ θ ≤ 360°
- Radian measure: 0 ≤ θ ≤ 2π
- Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1
Trigonometric Functions:
- Sine: sin(θ) = opposite side / hypotenuse
- Cosine: cos(θ) = adjacent side / hypotenuse
- Tangent: tan(θ) = opposite side / adjacent side
Trigonometric Identities:
- Sum and Difference Formulas
- Double and Half Angle Formulas

Coordinate Systems:
- Cartesian Coordinates (x, y)
- Polar Coordinates (r, θ)
Equations of Lines:
- Slope-Intercept Form: y = mx + b
- Point-Slope Form: y - y1 = m(x - x1)
- Standard Form: Ax + By = C
Circles and Conic Sections:
- Circle: (x - h)^2 + (y - k)^2 = r^2
- Ellipse: (x - h)^2/a^2 + (y - k)^2/b^2 = 1
- Parabola: y = a(x - h)^2 + k

Sequences:
- Arithmetic Sequence: an = a1 + (n - 1)d
- Geometric Sequence: an = a1 * r^(n - 1)
Series:
- Arithmetic Series: Σan = (a1 + an)/2 * n
- Geometric Series: Σan = a1 * (1 - r^n) / (1 - r)
Convergence Tests:
- nth Term Test
- Ratio Test
- Root Test

Domain of a function refers to the set of all input values (x) for which the function is defined.
Range of a function refers to the set of all output values (y) that the function can produce.
Function operations include sum/difference, product, and composition, with formulas (f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x) * g(x), and (f ∘ g)(x) = f(g(x)) respectively.
Functions can have even or odd properties, where f(-x) = f(x) or f(-x) = -f(x) respectively.
Functions can also be increasing or decreasing, where f(x) ≥ f(y) or f(x) ≤ f(y) for x > y.

Linear graphs are straight lines represented by the equation y = mx + b.
Quadratic graphs are parabolas represented by the equation y = ax^2 + bx + c.
Exponential graphs are represented by the equation y = a * b^x.
Trigonometric graphs are represented by the equations y = a * sin(x) or y = a * cos(x).
Graph transformations can be vertical shifts (y = f(x) ± k), horizontal shifts (y = f(x ± h)), or reflections (y = -f(x) or y = f(-x)).
Graphing techniques include plotting points, using symmetry, and identifying asymptotes.

Angles can be measured in degrees (0° ≤ θ ≤ 360°) or radians (0 ≤ θ ≤ 2π).
The Pythagorean Identity is sin^2(θ) + cos^2(θ) = 1.
Sine (sin), cosine (cos), and tangent (tan) are trigonometric functions, where sin(θ) = opposite side / hypotenuse, cos(θ) = adjacent side / hypotenuse, and tan(θ) = opposite side / adjacent side.
Trigonometric identities include sum and difference formulas, and double and half angle formulas.

Coordinate systems can be Cartesian (x, y) or polar (r, θ).
Equations of lines can be in slope-intercept form (y = mx + b), point-slope form (y - y1 = m(x - x1)), or standard form (Ax + By = C).
Circles can be represented by the equation (x - h)^2 + (y - k)^2 = r^2.
Ellipses and parabolas are types of conic sections, represented by the equations (x - h)^2/a^2 + (y - k)^2/b^2 = 1 and y = a(x - h)^2 + k respectively.

Arithmetic sequences have the form an = a1 + (n - 1)d, while geometric sequences have the form an = a1 * r^(n - 1).
Arithmetic series have the formula Σan = (a1 + an)/2 * n, while geometric series have the formula Σan = a1 * (1 - r^n) / (1 - r).
Convergence tests for series include the nth term test, ratio test, and root test.